2) Now assume that you also have the knowledge that $H = G^*G$ for some $G \in \mathcal{C}^{n \times m}$ with rank r. Does this piece of information reduce the number of real valued parameters needed to specify H?

3 Answers
3

The answer to 1) is simple. An Hermitean matrix is determined by its eigenvalues and
eigenspaces. Assign $r$ distinct eigenvalues ($r$ real parameters). Their
eigenspaces are orthogonal lines. One line depends on $n-1$ complex parameters, the next line
must be perpendicular to the first, so it depends on $n-2$ complex parameters, and so on.
Summing up, we obtain
$$r+2((n-1)+...+(n-r))=2nr-r^2$$
real parameters. To check, set $r=n$, we get $n^2$, the real dimension of the space of all
Hermitean matrices. Set $n=2,r=1$ we get $3$.

The answer to 2) is even simpler, because every Hermitean matrix can be represented in this
form.

1) Rank-$r$ Hermitian matrices are determined uniquely by their image $U$ and how they act when restricted to $U$. The image can be any dimension-$r$ subspace. Almost all subspaces have a basis in the form $\begin{bmatrix}I_r \\\\ X \end{bmatrix}$, with $X$ any $(n-r)\times r$ matrix, so you have $2(n-r)r$ degrees of freedom for the image. Possible actions on $U$ are isomorphic to $r\times r$ Hermitian matrices, so $r(r-1)+r=r^2$ real dof's. Overall this makes $2nr-r^2$ parameters.

2) If you know that $H$ is spd, then you have to restrict the second part to positive-definite matrices, but they still have the same number of parameters, so you still get the same answer.

The question seems a bit vague - what is a "parameter"? In one sense the answer is $r(n+1)$, since if $H$ is Hermitian, then $H=\sum_{i=1}^{r}{\lambda_{i}x_{i}x_{i}^{T}}$. Each $x_{i}$ has $n$ entries.